hydrogen atom in a magnetic ﬁeld: ghost orbits, catastrophes, … · 2018. 6. 13. · axis, vm n...

PHYSICAL REVIEW A MARCH 1997VOLUME 55, NUMBER 3

Hydrogen atom in a magnetic field: Ghost orbits, catastrophes,and uniform semiclassical approximations

Jorg Main* and Gunter WunnerInstitut fur Theoretische Physik I, Ruhr-Universita¨t Bochum, D-44780 Bochum, Germany

~Received 11 September 1996!

Applying closed-orbit theory to the recurrence spectra of the hydrogen atom in a magnetic field, one caninterpret most, but not all, structures semiclassically in terms of closed classical orbits. In particular, conven-tional closed-orbit theory fails near bifurcations of orbits where semiclassical amplitudes exhibit unphysicaldivergences. Here we analyze the role of ghost orbits living in complex phase space. The ghosts can explainresonance structures in the spectra of the hydrogen atom in a magnetic field at positions where no real orbitsexist. For three different types of catastrophes, viz. fold, cusp, and butterfly catastrophes, we construct uniformsemiclassical approximations and demonstrate that these solutions are completely determined by classicalparameters of the real orbits and complex ghosts.@S1050-2947~97!07002-9#

PACS number~s!: 32.60.1i, 03.65.Sq, 05.45.1b, 32.70.Cs

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I. INTRODUCTION

Rydberg atoms in external magnetic fields are nontrivsystems possessing a classically chaotic counterpart, wileast two nonseparable, strongly coupled degrees of freedEver since the discovery of quasi-Landau type modulatiin the spectra of barium@1# and hydrogen@2,3# atoms, andtheir interpretation in terms of classical periodic motion,oms in magnetic fields have served as prototype systemstudying quantum manifestations of classical chaos inphysical systems~for reviews see@4–6#!. As the correspond-ing classical dynamics of Rydberg atoms is chaotic,quantitative description of their quantum features in termsclassical orbits was, and still is, a big challenge to theoryregards the development and application of semiclassmethods.

A decisive advance for a semiclassical interpretationstructures in the photoabsorption cross section was achiby the development ofclosed-orbit theory@7,8#. The methodallows, at least in low resolution, not only a simple interptation but also a quantitative calculation of spectra in terof few parameters of the set of closed classical orbits starat and returning to the nucleus. Semiclassical results argood agreement with experimental data, e.g., of the hydgen atom@9#.

Closed-orbit theory fails, however, at energies wherebits undergo bifurcations, i.e., where closed or periodicbits are born or vanish. Bifurcation is a phenomenon typiof classical periodic orbits in chaotic systems with nonhypbolic Hamiltonian dynamics, such as the hydrogen atommagnetic fields, which undergoes a transition from regulato chaotic dynamics with increasing energy. The bifurcatscheme of this system has been analyzed in@10,11#. Nearbifurcations closed-orbit theory fails, the semiclassical fmulas diverge and are singular exactly at the bifurcatpoints. Such ‘‘catastrophes’’ not only occur in closed-or

*Present address: Dept. of Chemistry, University of SouthCalifornia, Los Angeles, CA 90089.

551050-2947/97/55~3!/1743~17!/$10.00

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theory but are well-known phenomena in various fieldsphysics, e.g., in semiclassical scattering theory@12,13#, dif-fraction theory in optics@14#, or periodic orbit theory@15#.Different kinds of catastrophes exist and are characterizedvarious forms of caustics of a bundle of lines~or, more spe-cific, in physical systems a bundle of classical trajectorieslight rays!. A systematic mathematical classification aanalysis of structurally stable caustics was achieved bydevelopment ofcatastrophe theory@16#.

The divergences in semiclassical theories can be remoby the construction ofuniform semiclassical solutions@12,13#. Their calculation is generally not unique becausetopological structure of the related catastrophe must be csidered. Uniform semiclassical approximations have bconstructed, e.g., for atomic and molecular scattering prlems@12,13,17,18#, photodetachment of H2 in parallel elec-tric and magnetic fields@19#, time-dependent wave-packepropagation@20#, and for continuum Stark spectra@21#. In anapplication of periodic orbit theory to a kicked top@22# itwas demonstrated that prebifurcation periodicghost orbitsexist, and are of importance in the semiclassical interpretion of that system.

It is the purpose of this paper to investigate the roleghost orbits and their relation to catastrophes and unifosemiclassical approximations in more detail. As a specsystem we study the hydrogen atom in a magnetic fielddemonstrate that around the singular points of standclosed-orbit theory, i.e., at the bifurcations of orbits, uniforsemiclassical approximations can be obtained from onlfew parameters of closed orbits, provided the type of catrophe is known, and not only real but also complex ghorbits are considered. The paper is organized as followsSec. II we discuss the classical dynamics and the contintion of closed orbits to complex phase space, i.e., the ghorbits. In Sec. III we derive uniform semiclassical appromations for three different kinds of catastrophes. A discsion of results and concluding remarks follow in Secs.and V.

II. CLASSICAL DYNAMICS AND GHOST ORBITS

The basic equations for the calculation of classical trajtories and a periodic orbit search for the hydrogen atom in

1743 © 1997 The American Physical Society

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1744 55JORG MAIN AND GUNTER WUNNER

magnetic field are given in many papers~see, e.g.,@4#!. Herewe briefly review the main ideas only, and concentrate onpeculiarities of a complex continuation of phase space,search for ghost orbits, and the calculation of the~complex!ghost orbit parameters.

The nonrelativistic Hamiltonian for the hydrogen atoma magnetic field of strengthB directed along thez axis hasthe well-known form @in atomic units, g5B/(2.353105 T)#

H51

2p22

1

r11

2gLz1

1

8g2%25E. ~2.1!

The component of the angular momentum parallel tofield axis is conserved and we chooseLz5m\50 in all clas-sical calculations.

A special feature of the Hamiltonian is its scaling proerty with respect to the magnetic-field strength. In scacoordinates and momenta,

r5g2/3r , p5g21/3p,

the classical Hamiltonian assumes the form

H5g22/3H51

2p22

1

r11

8%25E. ~2.2!

The classical trajectories obtained from the scaled equatof motion do not depend on both energy and magnetic-fistrength, but only on one parameter, the scaled eneE5Eg22/3. Note that the classical action scales as

S52pSg21/3. ~2.3!

The Coulomb singularity presents an obstacle to the numcal integration of the equations of motion that follow frothe Hamiltonian~2.2!. The way out of this problem is atransformation of timet°t, with dt52rdt, called regular-ization @23#, together with a coordinate transformationsemiparabolical coordinates

m5Ar1z, n5Ar2z. ~2.4!

These transformations lead to the regularized Hamiltonia

H5 12 ~pm

21pn2!2E~m21n2!1 1

8g2m2n2~m21n2!52,~2.5!

from which we obtain Hamilton’s equations of motion~theprimes denote derivativesd/dt)

m85pm , pm8 52Em2g2~m3n2/21mn4/4!,

n85pn , pn852En2g2~n3m2/21nm4/4!. ~2.6!

These equations are free of singularities, and were integrnumerically with the help of a high-order predictor-correcmultistep algorithm.

In a semiclassical approximation to photoabsorption sptra ~see Sec. III! closed orbits, which start at and return to tnucleus, are of fundamental importance. The closed-osearch can be formulated as finding roots (q i ,t f) of the twoequations

ee

e

d

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ri-

edr

c-

it

m~q i ,t f !5n~q i ,t f !50 ~2.7!

when integrating Hamilton’s equations~2.6! with the initialconditions~at t50)

m~0!50, pm~0!52cos~q i /2!

n~0!50, pn~0!52sin~q i /2!. ~2.8!

Here q i is the starting angle, i.e., the angle betweeninitial velocity of the electron and the field axis. Equatio~2.7! can be solved numerically with the help of an iteratiNewton algorithm, and its roots (q i ,t f) are thereal closedorbits @3,9# when all parameters are defined real.

The analytic structure of the equations of motion~2.6!allows a direct analytic continuation of the real phase sp(m,n,pm ,pn) to complex numbers. To search for compleclosed orbits we choose the same initial conditions~2.8! butwith complex starting angleq i , i.e., the momentapm(0) andpn(0) become complex, but the conditionH52 in Eq. ~2.5!is still satisfied. With these complex initial conditions Hamton’s equations of motion~2.6! can be integrated numericallfrom t50 to t5t f . It should be noted that in generalt falso must be chosen as a complex number to find ghost oas roots of Eq.~2.7!. In this case we integrate trajectoriealong straight lines in the complex plane fromt50 tot5t f , but the final parameters of ghost orbits such ascomplex recurrence time

T5E0

t f~m21n2!dt ~2.9!

and the action

S5E0

t f~pm

21pn2!dt ~2.10!

do not depend on any special choice of this path becausthe analytic structure of the equations of motion. This is atrue for the monodromy matrixM , i.e., the stability matrixrestricted to deviations perpendicular to a periodic orbit aperiodT. To be more specific, ifdq(0) is a small deviationperpendicular to the orbit in coordinate space at timet50anddp(0) an initial deviation in momentum space, the coresponding deviations at timet5T are related to the monodromy matrix@8#, viz.

S dq~T!

dp~T!D 5M S dq~0!

dp~0!D 5Sm11 m12

m21 m22D S dq~0!

dp~0!D .

~2.11!

It is not our intention in this paper to carry out a complesearch for all real and complex closed orbits. Rather,attention will be focused on the occurrence of ghost orbitsthe vicinity of bifurcations of real orbits, and we shall dicuss three specific examples, viz. the saddle node bifurcaof orbit X1, the period doubling ofV1

1, and a more complexbifurcation of the perpendicular orbitR1. Shapes of orbitsnear these bifurcations are shown in Fig. 1. For the clascation of closed orbits we adopt the nomenclature of@3#, i.e.,orbitsRm

n bifurcate from the motion perpendicular to the fie

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55 1745HYDROGEN ATOM IN A MAGNETIC FIELD: GHOST . . .

axis,Vmn denotes orbits bifurcating directly from the motio

parallel to the field, with higher-order bifurcations markedan asterisk, and orbitsXm are created ‘‘out of nowhere,’mostly in saddle-node bifurcations.

FIG. 1. Closed orbits~in semiparabolic coordinatesm,n) atscaled energies near bifurcation points.~a! Orbits X1

a and X1b at

energyE520.11. Both orbits are born in a saddle-node bifurcatat Eb520.115 442 16.~b! Ghost orbitX1 at energyE520.2.Solid line: Rem vs Ren. Dashed line: Imm vs Imn. ~c! Balloonorbit V1

1 at energyE520.32 ~solid line! and orbitV22* ~dashed

line! bifurcating in a period doubling atEb520.342 025 8.~d!Perpendicular orbitR2 ~solid line!, orbitR2

1 ~dashed line!, and orbitR21b ~dashed dotted line! at scaled energyE520.317.

A. Saddle-node bifurcation of the orbit X1

The first example we discuss in detail is the orbitX1,which is created in a saddle-node bifurcation at scaledergy Eb520.115 442 16 and with initial and final angleq i51.2271,q f52.4232 @10#. At higher energies (E.Eb)this orbit immediately splits up into two different real closeorbits (X1

a andX1b) with slightly different shapes, example

of which are shown in Fig. 1~a! for energyE520.11. Withdecreasing energy, both orbits are found to vanish exactlthe bifurcation point, and below the bifurcation ener(E,Eb) no real closed orbit with similar shape exists. However, the closed-orbit search extended to the complex ctinuation of phase space indeed reveals the existence ofbifurcation ghost orbits. For illustrational purposes the ghorbit is presented in Fig. 1~b! at energyE520.2. The realparts of semiparabolic coordinates~solid line! look similar toorbits found above the bifurcation energy@Fig. 1~a!# and theimaginary parts~dashed line! are usually relatively small.Note that the complex conjugate (q i* ,t f* ) of each ghost orbitis also a solution of Eq.~2.7!, i.e., there exist two closedorbits above and below the bifurcation energy which aregenerate exactly at the bifurcation point.

For the construction of uniform semiclassical approximtions in Sec. III some closed-orbit parameters, namely,initial and final angles, the classical action, and the elemm12 of the monodromy matrix, are of fundamental impotance. The energy dependence of these parameterscharacteristics around the bifurcation point which are relato the various types of bifurcations. In Fig. 2 we presentresults for the saddle-node bifurcation of the orbitsX1

a andX1b . The energy dependence of the starting anglesq i is given

in Fig. 2~a!. Both angles are real above the bifurcation e

i-

-

-

FIG. 2. ~a! Real and imaginarypart of starting angleq i for closedorbits around the saddle-node bfurcation of X1 at scaled energyEb520.115 442 16.G: Complexghost orbit. ~b! Difference inscaled actionDS56(S22S1)/2with S1,2 the action of the two realand ghost orbits respectively.~c!Real and imaginary part of monodromy matrix elementm12 forclosed orbits around the saddlenode bifurcation ofX1. Dashedlines: Analytical fits~see text!.

nt-


FIG. 3. ~a! Real and imaginarypart of starting angleq i for closedorbits around the period doublingbifurcation ofV1

1 at scaled energyEb520.342 025 8.G: Ghost or-bit. ~b! DifferenceDSbetween theclassical action of the~perioddoubled! balloon orbitV2

2 and realand ghost orbits bifurcating fromit. ~c! Monodromy matrix elementm12 of the balloon orbitV2

2 andorbits bifurcating from it. Al-though the ghost orbitG lives incomplex phase space its actioand monodromy matrix elemenremain real. Dashed lines: Analytical fits ~see text!.

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ergy, form a saddle at the bifurcation point, and becocomplex for the ghost orbits below the bifurcation energFigure 2~b! shows the deviation of the classical action frothe mean value,DS56(S22S1)/2, whereS1 andS2 are theclassical actions of the orbitsX1

a andX1b . Close to bifurca-

tion points the energy dependence of closed-orbit paramecan be approximated by analytical functions. ForDSwe ob-tain

DS56~ s/2p!~E2Eb!3/2 ~2.12!

with s54.96 @see the dashed line in Fig. 2~b!#. The mono-dromy matrix elementsm12 of both orbitsX1

a andX1b vanish

exactly at the bifurcation point, and can be approximatedE'Eb by

m1256M ~E2Eb!1/2 ~2.13!

with M5175, as is illustrated by the solid and dashed linin Fig. 2~c!. Note that the behavior of ghost orbit parameteDS and m12 at energiesE,Eb is simply the analytic con-tinuation of Eqs.~2.12! and~2.13!. Vice versa, a study of thebehavior of solely the real orbit parameters already suggthe existence of ghost orbits with properties revealed bymore sophisticated search for ghost orbits in complex phspace.

B. Period doubling of the balloon orbit V11

The second example of a bifurcation we study in detaithe period doubling of the balloon orbitV1

1 at scaled energyEb520.342 025 8. The balloon orbit itself is already crated at lower energyE520.3913 in a bifurcation from theorbit parallel to the field@10#. A special feature of its shape ithe symmetry in the initial and final angle, i.e.,q i5q f . Itexists below and above the period-doubling energy withany spectacular change of this shape. Above the perdoubling energy a new orbitV2

2* which breaks this symmetry (q iÞq f) separates fromV1

1 and is closed roughly aftetwo times the period ofV1

1. Examples of the shapes of thballoon orbit and its period-doubling bifurcationV2

2* areshown in Fig. 1~c!.

e.

rs

at

ss

tsese

s

td-

We searched for complex orbits in the vicinity of thbifurcation and found a ghost orbit~and its complex conju-gate! at energies below the bifurcation. The energy depdence of closed-orbit parametersq i , DS, and m12 is pre-sented in Fig. 3. There exist three real starting anglesq i atE.Eb , and one real and two complex angles atE,Eb @seeFig. 3~a!#. The classical action and the monodromy matelementm12 show a strange and unexpected behavior infollowing sense. Although the ghost orbit is embeddedcomplex phase space, these parameters remain exactlyeven far away from the bifurcation point. The differenDS in action between orbitsV2

2* andV22 is approximately

given by a parabola

DS5~ s/2p!~E2Eb!2 ~2.14!

with s518.27 @see Fig. 3~b!#, and the monodromy matrixelementsm12 of orbitsV2

2 andV22* are approximately linear

functions of the energy distance from the bifurcation poin

m1252M ~E2Eb! ~orbit V22!,

m1252M ~E2Eb! ~orbit V22* and ghost! ~2.15!

with M591 @see Fig. 3~c!#. Note that, as for the saddle-nodbifurcation~Sec. II A!, the energy dependence of ghost orparametersDS andm12 is simply the analytic continuation oEqs.~2.14! and~2.15! for the real orbitV2

2* , i.e., the param-eters remain real atE,Eb .

C. Bifurcation of the orbit perpendicular to the field

In this section we investigate real and ghost orbits relato the period doubling of the perpendicular orbitR1. Thisthird example of closed-orbit bifurcations is more compcated because various orbits with similar periods undetwo different elementary types of bifurcations at nearly tsame energy. The shapes of the real orbits at scaled enE520.317 are plotted in Fig. 1~d!, R1 ~solid linem5n) isthe orbit perpendicular to the magnetic-field axis, anddashed and dashed-dotted lines represent the orbitsR2

1 andR21b ~‘‘Pacmen’’ in @10#!.

-


FIG. 4. ~a! Real and imaginarypart of starting angleq i for closedorbits related to the bifurcatingscenario of the~period doubled!perpendicular orbitR2. ~b! Differ-enceDS between the classical action of the ~period doubled! per-pendicular orbitR2 and real andghost orbits bifurcating from it.~c! Monodromy matrix elementm12 of the perpendicular orbitR2

and orbits bifurcating from it.Dashed lines: Analytical fits~seetext!.

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und

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The structure of bifurcations and the appearance of ghorbits can be seen clearly in the energy dependence ostarting anglesq i in Fig. 4~a!. OrbitsR2

1 andR21b are created

in a saddle-node bifurcation atEb(1)520.317 353 45,

q i51.3465. Below the bifurcation energy we find~in anal-ogy to the saddle-node bifurcation discussed in Sec. II A! anassociated ghost orbit and its complex conjugate. OrbitR2

1b

is real only in a very short energy interval (DE'0.001), andis then involved in the next bifurcation aEb(2)520.316 185 37,q i5p/2. This is the period-doubling

bifurcation of the perpendicular orbitR1, which exists at allenergies@q i5p/2 in Fig. 4~a!#. The period doubling is simi-lar to the bifurcation ofV1

1 discussed in Sec. II B but with areversed energy dependence. The real orbitR2

1b separatesfrom R1 at energiesbelow the bifurcation point, i.e., a reaorbit vanishes with increasing energy. Consequently assated ghost orbits are expected at energiesabovethe bifurca-tion, i.e., E.Eb

(2) , and indeed such ‘‘postbifurcation’ghosts have been found. Its complex starting angles areshown in Fig. 4~a!.

The energy dependence of scaled actions, or, morecisely, the differenceDS with respect to the action of thperiod-doubled perpendicular orbitR2, is presented in Fig4~b! ~solid lines!, and the graph for the monodromy matrelementm12 is given in Fig. 4~c!. It can be seen that thactions and the monodromy matrix elements of the ghorbits related to the saddle-node bifurcation ofR2

1 becomecomplex atE,Eb

(1) , while these parameters remain real fthe postbifurcation ghosts atE.Eb

(2) . The two bifurcationsare so closely adjacent that neither Eqs.~2.12! and~2.13!, forthe saddle-node bifurcation, nor Eqs.~2.14! and ~2.15!, forthe period doubling, yield a reasonable approximationDS(E) andm12(E) in the neighborhood of the bifurcations

sthe

ci-

lso

e-

st

o

However, both functions can be fitted well by the more coplicated formulas

DS5k

2p„s~E2Eb

~2!!1 23 $16@s~E2Eb

~2!!11#3/2%…

~2.16!

and

m1252M ~E2Eb~2!! ~orbit R2!

m1254M ~E2Eb~2!!1

4M

s@16As~E2Eb

~2!!11#

~R21 , R2

1b , and ghosts! ~2.17!

with k53.76831024, s5763.6, andM513.52 @see thedashed lines in Figs. 4~b! and 4~c!#. Note that Eqs.~2.16! and~2.17! describe the complete scenario for the real andghost orbits including both the saddle-node and peridoubling bifurcations. We also mention that orbits wianglesq iÞq f have to be counted twice because they corspond to different orbits when traversed in either directioand therefore a total number offive closed orbits, includingghosts, is considered here in the bifurcaton scenario arothe period doubling of the perpendicular orbit.

III. UNIFORM SEMICLASSICAL APPROXIMATIONS

In this section we investigate in which way bifurcationsclassical orbits and the existence of ghost orbits manithemselves in quantum mechanics. More specificallyshall study the related quantum effects that can be obsein the photoabsorption spectra of the hydrogen atom in mnetic fields. The link between classical trajectories and p

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toabsorption spectra is established by a semiclassicalclosed-orbit theory@7,8#. In its original version, the elementm12 ofthe monodromy matrix appears in the denominator of seclassical expressions, and therefore the theory breaks dat the bifurcation points of orbits, wherem1250. In thefollowing we briefly review the general ideas of closed-ortheory and then derive uniform semiclassical approximativalid around the bifurcation energies of closed-orbit bifurctions. We demonstrate that the uniform solutions are direrelated to various types ofcatastrophes@16# formed by thebundle of returning trajectories and study three exampnamely, the fold, cusp, and butterfly catastrophe. The fiuniform expressions are free of singularities. In particuthe analysis will reveal the importance of classical ghostbits to quantum photoabsorption spectra.

A. The ansatz

The rationale of the semiclassical description of photosorption by atoms in strong magnetic fields is the followinAn electron in a low-lying initial stateuc i& is excited to aRydberg state, or a continuum state above the ionizathreshold. One finds a distancer 0 from the nucleus where thsemiclassical description of the wave functions becomegood approximation, but the Lorentz forces are still neglibly small compared to the Coulomb attraction forces. Byond r 0, the outgoing Coulomb wave describing the electrin the final state is propagated along classical trajectorThese trajectories obey the complete classical dynamicthe Hamiltonian~2.1!, i.e., with the effects of the magnetifield included. The latter become important at large distanfrom the nucleus, and the combined action of the magnefield forces and the Coulomb forces may cause trajectoriereturn to the nucleus. The returning waves interfere withinitial state in the dipole matrix element, and this interfeence gives rise to characteristic modulations in the photosorption cross section. Consequently, the semiclassicalpression for the oscillator strength is found to be compoof two parts, a smoothly varying continuous backgroundf 0

related to the initially excitedoutgoingCoulomb wave, andan oscillatory part from the contributions of allreturningwavescm

ret,k related to closed orbitsk (m designates the conserved magnetic quantum number!

f5 f 01 (cl.o.k

f kosc, ~3.1!

with

f kosc52

2

p~E2Ei !Im^c i uDucm

ret,k&. ~3.2!

The derivation of the semiclassical wave function has bdescribed in detail in the literature~see, e.g.,@7–9#!. Here weonly recapitulate the essential points that are necessary tunderstanding of what follows. The basic observation is tin the vicinity of the nucleus the orbits behave like reguKeplerian orbits, and every closed orbit~returning exactly tothe nucleus at an angleq f with respect to thez axis! will besurrounded by an ensemble of similar, almost-closed orbwhich approach the nucleus at the same angleq f , but swing

i-wn

ts-ly

s,al,r-

-:

n

a--ns.of

sc-toe-b-x-d

n

antr

s,

by the nucleus. For a given point (r ,q) in the neighborhoodof the nucleus, one can find an almost-closed orbit~associ-ated with a given closed orbitk) that passes through(r ,q). The almost-closed orbit contributes to the semiclascal wave function at (r ,q) with two terms, one belonging tothe ‘‘incoming’’ and one belonging to the ‘‘outgoing’’ trajectory along the almost-closed orbit@passage through(r ,q) before or after the perihelion, respectively#

cmret,k~r ,q!52A2psinq iYm~q i !

3 (l5out,in

expH i FSl,k~r ,q!2p

2ml,k1

p

4 G JAuJl,k~ t,q i !u

.

~3.3!

Hereq i is the starting angle of the closed orbit, and

Sl,k~r ,q!5Smk 1DSl,k~r ,q!, ~3.4!

ml,k5mk1Dml,k ~3.5!

(l 5 out, in! are the classical actions and Maslov indicesthe incoming and outgoing trajectory, respectively, where

Smk 5 R cl.o.k~pmdm1pndn!1mS 12 gTk1pnz,kD ~3.6!

denotes the classical action of the exactly closed orbitk in-cluding the action of the separablew motion. In Eq.~3.6!,Tk is the recurrence time, andnz,k the total number of crossings of the orbitk with the z axis. The Maslov indexmk

counts the total number of caustics along the closed orandDSl,k(r ,q) andDml,k represent the differences of thactions and Maslov indices of the incoming and outgotrajectory at the given point (r ,q) relative to the exactlyclosed orbitk. The quantityJl,k in Eq. ~3.3! is the Jacobideterminant of the incoming and outgoing trajectory,

Jl,k~ t,q i !5rsinqdetS ]~m,n!

]~t,q i !D l,k

. ~3.7!

The angular functionYm(q) in Eq. ~3.3! can be expressed interms of the matrix elements of the dipole operator withinitial stateuc i& and spherical harmonics~see Appendix A!.

The wave function~3.3! is the starting point for our construction of uniform semiclassical solutions. At values ofrwhere the semiclassical approximation is valid but thefects of the magnetic field can still be neglected, the seclassical returning wave~3.3! must be expandable in the basis of the exact quantum-mechanical Coulomb wafunctions. Taking for the latter their form atE'0 we canwrite

cmret,k~r ,q!5 (

l 5umu

`

cl mA2/rJ2l 11~A8r !Yl m~q,0!,

~3.8!

where the expansion coefficientscl m can be determinedfrom

es

hec

gp-g-

m

etuin

ourso-

nes

e

to

dng

theng

ard

l-


cl mA2/rJ2l 11~A8r !¥r→`

cl m

321/4p21/2r23/4~21! l cosSA8r23

4p D

52pE0

p

cmret,k~r ,q!Yl m~q,0!sinqdq

'2~2p!3/2Asinq isinq fYm~q i !Yl m~q f ,0!

3expS i FSmk 2p

2mk1

p

4 G D

3 (l5out,in

E0

pexpH i FDSl,k~r ,q!2

p

2Dml,kG J

F rUdetS ]~m,n!

]~t,q i !D l,kUdqG1/2 dq.

~3.9!

In Eq. ~3.9! we have exploited the asymptotic form of thBessel function, and the fact that the phase integral hamain contribution around the angleq'q f of the returningorbit, and that spherical harmonics with low-l quantumnumbers are smooth functions ofq. The form ~3.8! is cor-rect, in particular, in the region of the initial state, where tdipole matrix element has to be evaluated. Therefore weinsert Eq.~3.8! into Eq. ~3.2!, and obtain

f kosc52~E2Ei !Asinq isinq fYm~q i !Ym~q f !

3ImHA expS i FSmk 2p

2mk1

p

4 G D J , ~3.10!

where the complex amplitudeA is defined by

I ~r ![ (l5out,in

E0

pexpH i FDSl,k~r ,q!2

p

2Dml,kG J

FUdetS ]~m,n!

]~t,q i !D l,kUG1/2 dq

5AcosSA8r2

3

4p D

2p~2r !1/4~3.11!

~Eq. ~3.9! guarantees that the quantityI (r ) will always fac-torize in this form!. ThusA can be determined by evaluatinI (r ) at some value ofr where, again, the semiclassical aproximation is valid but magnetic-field effects are still neligible.

Equation~3.11! is the basis for the subsequent uniforapproximations of the amplitudeA. The decisive quantitiesentering intoA areDS, Dm, and the Jacobi matrix in theneighborhood of a returning~exactly closed! orbit. In thecase of bifurcations, i.e., catastrophes, the behavior of thquantities differs significantly from that in the standard siation of isolated returning orbits, as will become evidentthe following sections.

its

an

se-

B. The standard situation: Isolated returning orbits

For completeness, and illustrative purposes, we startdiscussion with the standard situation of nonbifurcating ilated returning orbits. We pick some closed orbitk, and willdrop the indexk for simplicity in what follows. In semipa-rabolic coordinates the returning trajectories are straight liin the vicinity of the origin, inclined at an angleq f /2 withrespect to them axis. It is most convenient to introduce throtated semiparabolic coordinates

j5mcosq f

21nsin

q f

25A2rcos

q2q f

2, ~3.12!

h5ncosq f

22msin

q f

25A2rsin

q2q f

2, ~3.13!

where thej andh axes are now parallel and perpendicularthe returning orbit ~see Fig. 5!. All orbits satisfy theHamilton-Jacobi equations

pj5]S

]j52, ph5

]S

]h50. ~3.14!

The central returning~i.e., exactly closed! orbit is given byh(j)50, while for the neighboring trajectories we have

h~j!5m12Dq i , ~3.15!

wherem12 is an element of the monodromy matrix, anDq i is the deviation of the starting angle of the neighboritrajectory. We immediately obtain

DSl~r ,q!562uju56A8rcosq2q f

2

56@A8r2Ar /8~q2q f !2#, q'q f ~3.16!

detS ]~m,n!

]~t,q i !D l

5detS ]~j,h!

]~t,q i !D l

52m12. ~3.17!

The Maslov index increases by 2 when the orbit passesorigin, i.e., the Maslov indices of the incoming and outgoitrajectory differ by

Dml561. ~3.18!

FIG. 5. Returning orbits close to the nucleus in the standsituation~monodromy matrix elementm12Þ0) in rotated semipara-bolic coordinates (j,h). Neighboring orbits are straight lines paralel to thej axis.

al

bi

ns,r

troi-

dcfo

ththe.hei-

r.

se

the-r,antral

e-anto-

n-n

it


Inserting Eqs.~3.16! to ~3.18! into Eq.~3.11! we obtain~withthe integration range of the stationary phase integral formextended to6`)

I ~r !5F 2

um12uG1/2E

2`

1`

sin@A8r2Ar /8~q2q f !2#dq

5Ap25/4r21/41

Aum12ucosSA8r2

3

4p D , ~3.19!

⇒A52~2p!3/21

Aum12u. ~3.20!

The contribution of a nonbifurcating isolated returning orto the oscillator strength then reads~with the scaled mono-dromy matrix elementm125g1/3m12)

f osc52~E2Ei !Asinq isinq fYm~q i !Ym~q f !2~2p!3/2g1/6

31

Aum12usinSSm0 2

p

2m01

p

4 D . ~3.21!

This is exactly the result of conventionalclosed-orbittheory@7,8#. Obviously the oscillator strength diverges, i.e., convetional closed-orbit theory fails, at bifurcations of orbitwherem1250. The reason for the occurrence of the divegence is that, in such a case, Eq.~3.15! does not correctlyrepresent the behavior in the neighborhood of the cenreturning orbit. Thus the decisive point in the constructiona uniform semiclassical approximation in the vicinity of bfurcating points is an adequate expansion of Eq.~3.15! tohigher orders inDq i . The order of the expansion requiredepends on the type of catastrophe, and therefore eachhas to be treated separately. We shall investigate thecusp, and butterfly catastrophes.

C. The fold catastrophe

An example for the occurrence of a fold catastophe inhydrogen atom in a magnetic field is the creation ofclosed orbitX1 through a saddle-node bifurcation at thscaled energyEb520.115 442 16 discussed in Sec. II AThe family of corresponding returning orbits satisfy tHamilton-Jacobi equations~in rotated semiparabolic coordnates, withpj5]S/]j andph5]S/]h)

cph21pj@jph2pj~h2h0!#50,

pj21ph

254 ~3.22!

from which we obtain~with ph /pj'aDq i)

h~j!5h01a2c~Dq i !21a~Dq i !j. ~3.23!

Hereh0, a, andc are constants which will be specified lateThe fold is illustrated in Fig. 6. At any point (j,h) Eq. ~3.23!is a quadratic equation inDq i , with the discriminant

D51

~2ac!2$j214c~h2h0!%. ~3.24!

ly

t

-

-

alf

aseld,

ee

If D.0 there exist two real solutions forDq i , i.e., twoorbits return to these points, but no orbit returns to thepoints ifD,0. The border line belonging toD50 betweenthe allowed and forbidden area is a caustic which hasshape of a parabola~see Fig. 6!. The Hamilton-Jacobi equations ~3.22! of the fold can be solved analytically. Howevein the following derivations it is more convenient to useapproximate solution valid in the neighborhood of the cenreturning orbit

DSl~r ,q!56FA8r2h0~q2q f !2c

12~q2q f !

3G ,q'q f . ~3.25!

The determinant in the denominator of Eq.~3.11! is found tobe given by

detS ]~m,n!

]~t,q i !D l

5detS ]~j,h!

]~t,q i !D l

562aAj214c~h2h0!¥r@0

6aA8r , q'q f ~3.26!

i.e., in the direction of the returning orbit it exhibits the bhavior of a focus of orbits at the origin. This focus causesadditional increase of the Maslov index of outgoing trajecries; i.e., we have

Dmout512, Dm in521. ~3.27!

We now insert Eqs.~3.25!, ~3.26!, and~3.27! into Eq.~3.11!.Extending formally again the integration range of the statioary phase integral to6`, we can express the integral iterms of the Airy function@26#:

FIG. 6. Fold catastrophe of returning orbits~in rotated semipa-rabolic coordinates! related to the saddle-node bifurcation of orbX1 at bifurcation energyEb520.115 442 16.

ci

mA

rtht

i-

he

n

torersyld

ior

lex

o-

of

a-ofhe.ost,theostam-vecalcalthes-s-

son-he


I ~r !521/4r21/4uau21/2

3expS 2 ip

4 D E2`

1`

cosSA8r23

4p2h0~q2q f !

2c

12~q2q f !

3Ddq

521/4r21/4uau21/2expS 2 ip

4 D cosSA8r23

4p D

32p~4/c!1/3Ai „~4/c!1/3h0… ~3.28!

⇒A525/2p2uau21/2

3expS 2 ip

4 D ~4/c!1/3Ai „~4/c!1/3h0…. ~3.29!

After inserting the amplitudeA into Eq. ~3.10!, the contri-bution of a fold catastrophe to the total semiclassical oslator strength~3.1! reads

f osc52~E2Ei !Asinq isinq fYm~q i !Ym~q f !25/2

3uau21/2~4/c!1/3Ai „~4/c!1/3h0…sinSSm0 2p

2m0D .

~3.30!

In Eq. ~3.30! the parametersa, c, andh0 are still undeter-mined. In the following we demonstrate that these paraeters are directly related to closed classical trajectories.can be seen from Eq.~3.24! no orbit returns to the origin(j5h50) for ch0.0 while two closed orbits exist foch0,0, i.e., at energies above the bifurcation energy. Inlimit ch0!0, therefore, Eq.~3.30! should lead to the resulof the standard situation~Eq. 3.21!, applied to the two closedorbits. Taking the asymptotic forms of the Airy function~seeAppendix B 1! in the limit ch0!0 we obtain from Eq.~3.30!

f osc52~E2Ei !Asinq isinq fYm~q i !Ym~q f !23/2p21/2

3uau21/2~2ch0/4!21/4

3FsinSSm0 22

3A24h0

3/c2p

2m01

p

4 D1sinSSm0 1

2

3A24h0

3/c2p

2~m011!1

p

4 D G .~3.31!

Comparing this expression with Eq.~3.21! we can identifythe contributions of two closed orbits with the Maslov indcesm0 andm011. The classical actionsS and monodromymatrix elementsm12 of these orbits, expressed in terms of tparametersa, c, andh0 in Eq. ~3.31!, must be identical tothose obtained directly from classical trajectory calculatio~see Sec. II A!, from which we find the relations

S5Sm0 7 2

3A24h03/c[Sm

0 7g21/3s~E2Eb!3/2,

~3.32!

l-

-s

e

s

m12562p4ac2/3A2h0c21/3[6g21/3M ~E2Eb!

1/2.~3.33!

With the help of these relations the semiclassical oscillastrength~3.30! can be expressed in terms of the paramets and M that follow directly from the classical trajectorcalculations. The final result for the contribution of a focatastrophe thus reads

f osc52~E2Ei !Asinq isinq fYm~q i !Ym~q f !27/2

3p2g1/9~3s/2!1/6uM u21/2

3Ai „~3s/2!2/3g22/9~Eb2E!…

3sinSSm0 2p

2m0D . ~3.34!

It is also illustrative to investigate the asymptotic behavof Eqs.~3.30! and~3.34! in the limit ch0@0; i.e., at energiesE below the bifurcation energyEb , where no orbit returns tothe origin, because this reveals the role of the comp‘‘ghost’’ orbits discussed in Sec. B 1. In the caseE!Eb theasymptotic form of the Airy function~see Appendix B 1! canbe used in Eq.~3.34! to obtain

f osc52~E2Ei !Asinq isinq fYm~q i !Ym~q f !2~2p!3/2

3g1/6uM u21/2~Eb2E!21/4

3ImH expS i FSm0 1 i sg21/3~Eb2E!3/22p

2m0G D J .

~3.35!

Formally this equation coincides with Eq.~3.21!, but for aclosed orbit with a complex action and imaginary mondromy matrix element

S5Sm0 1 ig21/3s~Eb2E!3/2, ~3.36!

m125 ig21/3M ~Eb2E!1/2. ~3.37!

This is precisely the behavior found in the investigationghost orbits at energies below a saddle-node bifurcation~seeSec. II A!. Obviously, as one moves away from the bifurction point to smaller energies, the positive imaginary partS grows, and thus the contribution of the ghost orbit to tsemiclassical oscillator strength is damped exponentially

In the classical calculations, the complex-conjugate ghwith negative imaginary part of the action, also exists. Inabove semiclassical formulas this complex-conjugate ghwould produce an unphysical exponential increase of theplitude at energies below the bifurcation point. Thus we haas a by-product of the derivation of uniform semiclassiformulas that ghost orbits of this type have no physimeaning. In other words, they must not be included instandard formula~3.21! since they do not appear in the aymptotic expansion of the uniform approximation of the ocillator strength.

The uniform solution~3.34! for the fold catastrophe hathe benefit of connecting both asymptotic situations, the ctribution of the physical ghost orbit at energies below t

ieat

t ohe

ng

7

elu

a

t i

d

q.

antotic

he

e


saddle-node bifurcation and the two real orbits at energabove the bifurcation, without exhibiting any singularitythe bifurcation point.

D. The cusp catastrophe

The second fundamental catastrophe is the cusp, as icurs, for example, at the period-doubling bifurcation of tballoon orbit V1

1 at scaled energyEb520.342 025 8~seeSec. II B!. The Hamilton-Jacobi equations for the returniorbits at the cusp read~with pj5]S/]j andph5]S/]h)

2cph31pj

2@~j2j0!ph2hpj#50,

pj21ph

254, ~3.38!

and we obtain~with ph /pj'aDq i and parametersa, c, andj0 specified later!

h~j!52a3c~Dq i !31a~Dq i !~j2j0!. ~3.39!

The bunch of orbits forming the cusp is shown in Fig.Equation ~3.39! is a cubic polynomial inDq i with one orthree real zeros, depending on the sign of

D527ch212~j2j0!3. ~3.40!

Both regions are separated by a caustic given byD50, i.e.,h25(2/27c)(j02j)3. In contrast to the fold catastrophthere is no classical forbidden region without any real sotion of Eq. ~3.39!.

We shall now derive a uniform semiclassical approximtion for the cusp catastrophe. The decisive point is againfind a solution for the classical action and the determinanEq. ~3.11!. Solving the Hamilton-Jacobi equations~3.38! inthe vicinity of the central returning orbit (r@0, q'q f) weobtain the classical action

FIG. 7. Cusp catastrophe of returning orbits~in rotated semipa-rabolic coordinates! related to the period-doubling bifurcation of thballoon orbitV1

1 at bifurcation energyEb520.342 025 8.

s

c-

.

-

-ton

DSl~r ,q!56A8r1 14 j0~q2q f !

22 116c~q2q f !

4

~3.41!

and for the determinant in the denominator of Eq.~3.11! wehave

detS ]~m,n!

]~t,q i !D l

5detS ]~j,h!

]~t,q i !D l

¥

r@06aA8r , q'q f .

~3.42!

After summing up the contributions of the incoming anoutgoing orbit ~with the correct Maslov indicesDmout512,Dm in521), the stationary phase integral in E~3.11! for the cusp catastrophe reads~with t[q2q f)

I ~r !521/4r21/4uau21/2expS 2 ip

4 D cosSA8r23

4p D

3E2`

1`

exp$ i @~j0/4!t22~c/16!t4#%dt. ~3.43!

From Eq.~3.11! we now obtain the amplitude

A525/2puau21/2expS 2 ip

4 D c21/4F~2c21/2j0!,

~3.44!

where

F~x![E2`

1`

exp~2 i @xt21t4# !dt ~3.45!

is a special case of Pearcey’s integral@25# that can be solvedanalytically ~see Appendix B 2!. Inserting the amplitudeA@Eq. ~3.44!# into Eq.~3.10! we finally obtain the contributionof the cusp to the total oscillator strength~3.1!

f osc52~E2Ei !Asinq isinq fYm~q i !Ym~q f !25/2p

3uau21/2c21/4ImH expS i FSm0 2p

2m0G DF~2c21/2j0!J .

~3.46!

To find the relation between the parametersa, c, andj0 andparameters of the closed classical orbits we proceed inanalogous manner as for the fold, and discuss the asympbehavior of Eq.~3.46! in the limit j0!0 andj0@0, i.e., atscaled energies far from the bifurcation point. Applying tasymptotic formulas forF(x) @Eq. ~B9! of Appendix B 2#we obtain

f osc52~E2Ei !Asinq isinq fYm~q i !Ym~q f !2~2p!3/2uaj0u21/2

3H sinFSm0 2p

2~m011!1

p

4 G ; j0!0

sinSSm0 2p

2m01

p

4 D 1A2sinFSm0 1j02

4c2

p

2~m011!1

p

4 G ; j0@0.

~3.47!

-

inoe

of

d,bth

ath

twto

a

w

e

brg

tr

aher-bebE

dor-nshe,

by

m-,

eobi-

-

five


By comparing with Eq.~3.21! and with the classical dynamics at the period-doubling bifurcation~see Sec. II B! we findthe following physical interpretation of Eq.~3.47!. There isone real closed orbit with classical actionSm

0 which existsboth below and above the bifurcation point. The Maslovdex of this orbit changes by one when the orbit undergthe bifurcation and the monodromy matrix elementm12 mustobey the relation

m1252aj0[2g21/3M ~E2Eb!. ~3.48!

The remaining term atj0@0 can be interpreted as the sumtwo real orbit contributions with the Maslov indexm011and classical action and monodromy matrix element

S5Sm0 1

1

4cj02[Sm

0 1g21/3s~E2Eb!2, ~3.49!

m1252aj0[2g21/3M ~E2Eb!. ~3.50!

In the investigation of closed orbits around the periodoubling bifurcation~Sec. II B!, in addition to the real orbitstwo ghost orbits have been found at energies below thefurcation energy with the somewhat strange property thatclassical actionS and monodromy matrix elementm12 bothremain real, although the coordinates and momenta in phspace are complex. Contrary to the fold catastrophe, inasymptotic expansion~3.47! of the uniform solution~3.46!for a cusp, no ghost contribution appears; i.e., theseghost orbits are both without physical meaning in the phoabsorption process. With the help of Eqs.~3.49! and ~3.50!we can finally express the uniform solution~3.46! com-pletely in terms which are accessible from closed-orbit cculations

f osc52~E2Ei !Asinq isinq fYm~q i !Ym~q f !8pg1/12s1/4

3uM u21/2 ImH expS i FSm0 2p

2m0G D

3F@2s1/2g21/6~Eb2E!#J . ~3.51!

E. The butterfly catastrophe

As an example of even more complicated catastrophesinvestigate in this section the bifurcation ofR2, i.e., the sec-ond return of the perpendicular orbit around the scaledergyE520.316. A detailed classical analysis~see Sec. II C!exhibits a sequence of two different elementary types offurcations at almost the same point. At the scaled eneEb(1)520.317 353 45 two closed orbits, namely,R2

1 and asecond orbit with similar shape, which we callR2

1b, are cre-ated by a saddle-node bifurcation related to a fold catasphe. At the only slightly higher energy Eb

(2)5

20.316 185 37 the orbitR21b already vanishes again in

period-doubling bifurcation with the second return of tperpendicular orbitR2, i.e., in a catastrophe where neighboing returning orbits form a cusp. The energy spacingtween both bifurcations is so small that they cannottreated as isolated, and none of the uniform formulas

-s

-

i-e

see

o-

l-

e

n-

i-y

o-

-eq.

~3.34! for the fold and Eq.~3.51! for the cusp, can be appliein a semiclassical calculation of oscillator strengths. The crect uniform approximation must describe both bifurcatiosimultaneously, i.e., a more complicated type of catastropviz. a butterfly catastrophe, must be investigated.

The bunch of trajectories forming the butterfly is giventhe Hamilton-Jacobi equations~with pj5]S/]j andph5]S/]h)

3dph512cpj

2ph31pj

4@~j2j0!ph2hpj#50,

pj21ph

254, ~3.52!

where the parametersc, d, and j0 will be specified later.With ph /pj'aDq i we obtain

h~j!53a5d~Dq i !512a3c~Dq i !

31a~Dq i !~j2j0!.~3.53!

The butterfly is illustrated in Fig. 8. Depending on the nuber of real solutionsDq i of Eq. ~3.53! there exist one, threeor five orbits returning to each point (j,h). The differentregions are separated by caustics.

To find a uniform semiclassical approximation for thbutterfly catastrophe, we have to solve the Hamilton-Jacequations~3.52!, at least in the vicinity of the central returning orbit, and to insert the actionS(r ,q) into Eq.~3.11!. Forthe classical action we obtain

FIG. 8. ~a! Butterfly catastrophe of returning orbits~in rotatedsemiparabolic coordinates! related to the bifurcation of the perpendicular orbit at Eb520.316 185 37. ~b! Magnification of themarked region close to the nucleus. There are one, three, ororbits returning to each point (j,h) in phase space.

l

dof

hed,

laohe

-

rbit

itcal

ed

-m-

am-

is-Atndandts do

psitu-


DSl~r ,q!56A8r1 14 j0~q2q f !

21 116c~q2q f !

4

2 164d~q2q f !

6, ~3.54!

and for the determinant in the denominator of Eq.~3.11! wefind 6aA8r in the limit r@0 andq'q f , i.e., the sameresult ~3.42! as for the cusp. Summing up in Eq.~3.11! thecontributions of the incoming and the outgoing orbit~withMaslov indicesDml as for the cusp! we obtain the integraI (r ) and the amplitudeA for the butterfly catastrophe(t[q2q f)

I ~r !521/4r21/4uau21/2expS 2 ip

4 D cosSA8r23

4p D

3E2`

1`

exp$ i @~j0/4!t22~c/16!t42~d/64!t6#%dt,

~3.55!

⇒A525/2puau21/2expS 2 ip

4 Dd21/6C~2d21/3j0 ,2cd22/3!,

~3.56!

where

C~x,y![E2`

1`

exp~2 i @xt21yt41t6# !dt ~3.57!

is an analytic function in both variablesx andy. Its numeri-cal calculation and asymptotic properties are discusseAppendix B 3. The uniform result for the oscillatory partthe transition strength now reads

f osc52~E2Ei !Asinq isinq fYm~q i !Ym~q f !25/2p

3uau21/2d21/6

3ImH expS i FSm0 2p

2m0G DC~2d21/3j0 ,2cd22/3!J .

~3.58!

It is very illustrative to study the asymptotic behavior of tuniform approximation~3.58! as we obtain, on the one hanthe relation between the parametersa, c, d, andj0 and theactions and the monodromy matrix elements of closed csical orbits, and, on the other hand, the role of complex ghorbits related to this type of bifurcation is revealed. In tfollowing we discuss both limitsj0@0, i.e., scaled energyE@Eb

(2) , andj0!0, i.e., E!Eb(1) .

1. Asymptotic behavior at scaled energy E˜@Eb„2…

Applying Eq.~B13! from Appendix B 3 to theC functionin the uniform approximation~3.58!, we obtain the asymptotic formula forj0@0

in

s-st


3uaj0u21/2H sinSSm0 2p

2m01

p

4 D1F11

c2

3dj0@11A~3d/c2!j011#G21/2

3sinSSm0 1c3

9d2 H 3dc2 j012

3F11S 113d

c2j0D 3/2G J

2p

2~m011!1

p

4 D J . ~3.59!

Comparing this with Eq.~3.21! we can identify the contribu-tions of three real closed orbits. The classical action of o1 is Sm

0 , its Maslov index ism0 and the monodromy matrixelementm12 is given by

m12~1!52aj0[2g21/3M ~E2Eb!, ~3.60!

where the parameterM can be determined by closed-orbcalculations~see Sec. II C!. Orbits 2 and 3 are symmetriwith respect to thez50 plane and have the same orbitparameters, i.e., Maslov indexm011 and the classical actionand the monodromy matrix element

S~2,3!5Sm0 1DS

5Sm0 1

c3

9d2 H 3dc2 j012

3F11S 3dc2 j011D 3/2G J ,~3.61!

m12~2,3!54aj0F11

c2

3dj0@11A~3d/c2!j011#G .

~3.62!

In the example of the bifurcation of orbitR2 at scaled energyEb(2)520.316 185 37, orbit 1 is the orbitR2 perpendicular to

the magnetic-field axis, while orbits 2 and 3 can be identifiwith R2

1 traversed in both directions (q i ,35p2q i ,2). Withthe help of Eqs.~3.60! to ~3.62! and classical scaling properties of the action and the monodromy matrix the paraetersa, c, d, andj0 in the uniform approximation~3.58! cannow be expressed completely in terms of closed-orbit paretersk, s, andM ~see Sec. II C!,

uau21/2d21/6531/6g1/18k1/3~ s/M !1/2,

d21/3j0531/3k2/3g22/9s~E2Eb~2!!, ~3.63!

cd22/35~9k!1/3g21/9.

In the classical analysis complex ghost orbits were dcovered both below and above the bifurcation energy.E.Eb

(2) they have the property that the classical action athe monodromy matrix remain real, although coordinatesmomenta in phase space are complex. These ghost orbinot appear in the asymptotic expansion~3.59! of the uniformapproximation~3.58!, and therefore, in analogy with the cuscatastrophe, they do not have a physical meaning. The

an

itley

ith

othdecavayicexaeq

uer

idf itiasr

or-n aultsarder-eree ofical

-

tate

l

t inre-rgy

f

-e-ntialitsartn-des-


ation is different at energyE,Eb(1) where a ‘‘hidden ghost’’

with physical meaning will be revealed in Sec. III E 2.

2. Asymptotic behavior and ‘‘hidden ghost’’at scaled energy E˜!Eb

„1…

At scaled energies below the bifurcation point we capply the asymptotic formula~B15! from Appendix B 3 totheC function in the uniform approximation~3.58! and ob-tain


3uaj0u21/2H sinSSm0 2p

2~m011!1

p

4 D1ImH F11

c2

3dj0@12 iA2~3d/c2!j021#G21/2

3expS i FSm0 1c3

9d2 H 3dc2 j0

12

3F11 i S 23d

c2j021D 3/2G J 2

p

2m01

p

4 G D J J .~3.64!

The first term in Eq.~3.64! can be identified as a real orbwith the same classical action and monodromy matrix ement as in Eq.~3.59!, but with a Maslov index increased bone. The second term in Eq.~3.64! is a ghost orbit contribu-tion resulting from a superposition of two closed orbits wcomplex action and monodromy matrix element,

S~2,3!5Sm0 1

c3

9d2 H 3dc2 j012

3F11 i S 23d

c2j021D 3/2G J ,

~3.65!

m12~2,3!54aj0F11

c2

3dj0@12 iA2~3d/c2!j021#G ,

~3.66!

traversed in both directions. The positive imaginary partthe classical action results in an exponential damping ofghost orbit contribution to the oscillator strength amplituwith decreasing energy similar to the situation at a foldtastrophe discussed above. In contrast to the fold, howethe ghost orbit related to a butterfly catastrophe is alwaccompanied by a real orbit with almost the same classaction. Because the contribution of the real orbit is notponentially damped its amplitude at energies where theymptotic formula~3.64! is valid is much stronger than thghost contribution. Therefore we call the ghost orbit in E~3.64! a ‘‘hidden ghost,’’ and it might be rather difficult tofind evidence for such a hidden ghost orbit related to a bterfly catastrophe in the Fourier transform of, e.g., expmental scaled energy spectra.

Note that, classically, the complex conjugate of the hden ghost orbit also exists. The negative imaginary part oclassical action would result in an unphysical exponenincrease of amplitude with decreasing energy, and conquently the complex-conjugate ghost orbit does not appeathe asymptotic formula~3.64!.

-

fe

-er,sal-s-

.

t-i-

-tsle-in

Finally, after inserting Eq.~3.63! in Eq. ~3.58!, the uni-form approximation for the butterfly catastrophe reads

f osc52~E2Ei !Asinq isinq fYm~q i !Ym~q f !

3pg1/1831/6k1/3~32s/M !1/2

3ImH expS i FSm0 2p

2m0G D

3C„231/3k2/3g22/9s~E2Eb~2!!,2~9k!1/3g21/9

…J .~3.67!

IV. RESULTS AND DISCUSSION

We now discuss the effects of bifurcations and ghostbits on photoabsorption spectra of the hydrogen atom imagnetic field and compare the uniform semiclassical resderived in the preceding chapter to solutions of standclosed-orbit theory. Semiclassically each closed orbit genates a modulation in the photoabsorption cross section. Hwe are not interested in the energy and field dependencthese modulations, which are directly related to the classactionS of closed orbits, but in the behavior of theirampli-tudes, especially in the vicinity of bifurcations. In the following we also drop the common prefactor:

2~E2Ei !~sinq isinq f !1/2Ym~q i !Ym~q f !

in formulas for the oscillatory partf osc of the photoabsorp-tion cross section. This prefactor depends on the inital sand excitation process~see Appendix A! and is, for low-lying initial states, a slowly varying function of the initiaand final angles of closed orbits. The amplitudeA(E,g) cannow be defined as

A~E,g!5U(kAk~E,g!expS i FSmk ~E,g!2

p

2mk1

p

4 G DU.~4.1!

By taking the absolute value instead of the imaginary parEq. ~4.1! we suppress, for the graphical presentation ofsults, the high oscillatory modulations caused by the eneand field dependence of the classical actionSm

k (E,g). Nev-ertheless, the exponential function in Eq.~4.1! is importantwhen applying Eq.~4.1! to nonuniform standard solutions oclosed-orbit theory with coefficientsAk given by Eq.~3.20!,i.e., ~in scaled variables!

Ak52~2p!3/2g1/6@m12,k~E!#21/2. ~4.2!

In this case we add, in Eq.~4.1!, the contributions of all realclosed orbits and ghostsk which are included in the corresponding uniform approximation. Interference effects btween these orbits are now taken care of by the exponefunction. Furthermore, the classical action of ghost orbmay be complex, which results, for a positive imaginary pof the action, in an exponential damping of ghost orbit cotributions. Note that the energy dependence of amplitu~4.1! can be measured experimentally byscaled energy spec

-n

d

w

erthbononhothe-inanine

lc-

ry

ut,wli-

iod-

ur-on,

fthetedere-on

udeed

rsn-ral,ewsesn-

ither-inofen-

herflyce-er-

thedesthe


troscopy @3,9,21# when following the peak hights of resonances in the (E,S) diagram of Fourier transform actiospectra.

First we discuss the fold catastrophe related to the sadnode bifurcation of orbitX1. The amplitudesA(E,g) at con-stant magnetic-field strengthg5431026 are presented inFig. 9~a!. The dashed line shows the superposition of the treal orbitsX1

a and X1b at energiesE.Eb . The oscillatory

structure of the amplitude is the result of a strong interfence between these two orbits. At the bifurcation energyamplitude diverges. Note that the amplitude at energieslow the bifurcation point is zero in the standard formulatiof the closed-orbit theory, i.e., when only real orbits are csidered. The dashed-dotted line is the extension when gorbit contributions are also included. We only considerghost with a positive imaginary part of complex action bcause only this ghost has a physical meaning~see the discussion in Sec. III C!. The amplitude decreases exponentially2(Eb2E)3/2 with decreasing energy, but also exhibitsunphysical divergence at the bifurcation point. The solid lin Fig. 9~a! is the uniform approximation for the amplitudwhich we obtain from Eq.~3.34! as

A~E,g!527/2p2g1/9~3s/2!1/6uM u21/2

3uAi „~3s/2!2/3g22/9~Eb2E!…u, ~4.3!

with parametersEb , s, andM determined from the classicatrajectory calculations in Sec. II A. Apart from constant fa

FIG. 9. Semiclassical amplitudes.~a! X1 ~fold catastrophe!,magnetic field strengthg5431026 a.u.;~b! V2

2 ~cusp catastrophe!,g51027; ~c! R2 ~butterfly catastrophe!, g51027. Dashed lines:Amplitudes in the standard model. Dashed-dotted line@in ~a!#:Ghost orbit contribution. Solid lines: Uniform approximations.

le-

o

-ee-

-ste-

e

tors, the uniform approximation to the amplitude is an Aifunction in terms of the energy differenceEb2E. Both theoscillatory structure of the amplitude atE.Eb and the ex-ponentially damped ghost orbit tail are well reproduced, bin addition, the singularity at the bifurcation point is nocompletely removed. Note that the maximum of the amptude is not located at the bifurcation pointE5Eb but isshifted to a slightly higher energy.

The results for the cusp catastrophe related to the perdoubling bifurcation of the balloon orbitV1

1 are illustrated inFig. 9~b! ~magnetic-field strengthg51027). The nonuni-form amplitude is plotted as a dashed line. Above the bifcation energy, three real orbits, the period-doubled balloV22, and the bifurcated orbit,V2

2* , traversed in both direc-tions, are considered in Eq.~4.1!, and the interference othese three orbits produces the oscillatory fluctuations ofamplitude. As discussed in Sec. III D the ghost orbits relato the cusp catastrophe have no physical meaning and, thfore, only the real orbit, i.e., the period doubled balloV22, is considered atE,Eb . The nonuniform solution is

characterized by an unphysical divergence of the amplitaround the bifurcation energy, but the singularity is removin the uniform approximation@solid line in Fig. 9~b!#

A~E,g!58pg1/12s1/4uM u21/2uF„2s1/2g21/6~Eb2E!…u~4.4!

obtained from Eq.~3.51!, and with the classical parameteEb , s, and M determined in Sec. II B. The energy depedence is given by a special case of Pearcey’s integF(x) ~see Appendix B 2!. In contrast to the fold catastroph@Fig. 9~a!# there is a real orbit, instead of a ghost orbit, belothe bifurcation point and therefore the amplitude decrea;(Eb2E)21/2 rather than exponentially with decreasing eergy. The modulations of amplitudes atE.Eb are also lesspronounced, and, in particular, there are no energies wvanishing amplitude, i.e., with complete destructive interfence between all closed-orbit contributions. Oscillationsperiodic orbit contributions related to a cusp catastrophereturning trajectories have recently been verified experimtally in Stark spectra of lithium atoms@21#.

Finally we discuss the amplitudes of modulations for tbutterfly catastrophe of returning trajectories. The buttecatastrophe is related to a more complicated bifurcation snario as found, e.g., around the period doubling of the ppendicular orbit~see Sec. II C!. The uniform approximationto the amplitudes is obtained from Eq.~3.67!,

A~E,g!5pg1/1831/6k1/3~32s/M !1/2

3uC@231/3k2/3g22/9s~E2Eb~2!!,

2~9k!1/3g21/9#u, ~4.5!

with the closed-orbit parametersk, s, M , and Eb(2) deter-

mined in Sec. II C and the integralC(x,y) solved in Appen-dix B 3. Results forg51027 are presented in Fig. 9~c!. Thesolid line is the uniform approximation~4.5!, and for com-parison the dashed line shows the nonuniform solution ofstandard closed-orbit theory. The modulations of amplituat energies above the bifurcation point are caused by

od

es

epetthe

elbi

rmvetie

nd-a-

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altin

iinu

ieve

asurit

enodesl

rfl

xi-the

er isen-phe

theisa-alelat-ich

nsonsof

r ofinpectn-of

nsas

gford.

l-

i-

-


interference of three closed orbits, namely, the peridoubled perpendicular orbit,R2, and the ‘‘Pacman’’ orbit,R21, traversed in both directions. Ghost orbits exist at th

energies but have no physical meaning~see Sec. III E!. Be-low the bifurcation point one real orbit,R2, and, in addition,a ghost orbit~and its time reversed counterpart! contribute tothe semiclassical photoabsorption spectrum. As in the casa fold catastrophe the amplitude of the ghost orbit is damexponentially with decreasing energy, but now the ghosnot isolated and its contribution is surpassed by that ofperpendicular orbit, i.e., the ghost is ‘‘hidden’’ behind threal orbit.

V. CONCLUSIONS

Photoabsorption spectra of hydrogen in a magnetic ficalculated semiclassically by application of closed-ortheory in its original version@7,8#, suffer from singularitiesat energies where orbits undergo bifurcations. The unifoapproximations derived in this paper significantly improon those calculations and remove unphysical singularifrom the spectra. The total oscillator strength~3.1! is dividedinto three parts,

f5 f 01 (kstandard

f kstandardosc 1 (

k uniformf k uniformosc , ~5.1!

with f 0 the smoothly varying continous background af k,standardosc the oscillatory part from the contributions of isolated closed orbits sufficiently far away from any bifurctions. In this case the returning trajectories do not formcatastrophe at the origin and Eq.~3.21! can be applied. Thethird term in Eq.~5.1! adds the contributions of closed orbinear bifurcations, i.e., when returning trajectories form atastrophe at the origin. The uniform approximations depeon the specific type of the catastrophe. In this paper we hinvestigated, in detail, the fold, cusp, and butterfly catasphe, wheref k,uniform

osc is given by Eqs.~3.34!, ~3.51!, and~3.58!, respectively. Note that each of these terms usurepresents the contributions of several strongly interacclosed orbits including ghosts.

We believe that a large variety of singularities in semclassical photoabsorption spectra of the hydrogen atommagnetic field can be removed by one of the uniform formlas mentioned above. The ‘‘exotic’’ orbits@3,9# are usuallycreated in a saddle-node bifurcation, with the trajectorforming a fold close to the origin. As an example we hastudied the shortest exotic orbitX1, but obviously the uni-form result for the fold catastrophe can be applied inanalogous situations. Also the uniform solution for the cucatastrophe is not at all restricted to a period-doubling bifcation of orbits~e.g., the period doubling of the balloon orbV11) but is valid for any period-n bifurcation whenever re-

turning orbits form a cusp. Last, but not least, the perpdicular orbitR1 undergoes an organized sequence of perin bifurcations at energiesE,20.1273, at which energy thorbit finally becomes unstable. The systematics of thisquence is explained in@11# by application of the normaform theory. The creation and annihilation of orbitsRm

n issimilar to the bifurcation scenario of orbitsR2 andR2

1 dis-cussed in this paper, the returning orbits form a butte

-

e

ofdise

d,t

s

a

-dve-

lyg

-a-

s

llp-

--

e-

y

close to the origin, and the corresponding uniform appromations can be applied to semiclassical calculations ofphotoabsorption cross section.

However, the set of catastrophes discussed in this papcertainly not complete. The possible geometries of elemtary catastrophes are systematically studied in catastrotheory~see, e.g.,@16#!. A complete investigation of all typesof catastrophes existing in atoms in external fields andderivation of related uniform semiclassical approximationssubject to future work. For example, closed-orbit bifurctions might exist where the returning trajectories formswallowtail. We also do not study bifurcations of the paralorbit in this paper. The parallel orbit requires special trement because of a symmetry property of nearby orbits whare invariant under rotations of the azimuthal angle,w.

The application of uniform semiclassical approximatiomust not be restricted to the hydrogen atom. Investigatiof nonhydrogenic atoms have revealed the importanceclassical trajectories scattered at the ionic core@24#. The corescattering results in a dramatic increase in the numbeclosed orbit bifurcations which are related to singularitiesthe semiclassical photoabsorption spectra. Thus we exuniform approximations to be a powerful tool to remove uphysical singularities from semiclassical spectra, not onlyhydrogen but of nonhydrogenic atoms as well.

ACKNOWLEDGMENTS

We thank Professor F. Haake for stimulating discussioand for bringing ghost orbits to our attention. This work wsupported by the Deutsche Forschungsgemeinschaft~SFB237!. J.M. is grateful to Alexander von Humboldt-Stiftunfor financial assistance, and to Professor Howard Taylorhis kind hospitality at USC, where this work was complete

APPENDIX A: ANGULAR FUNCTION Ym„q…

The angular functionYm(q) solely depends on the initiastatec i and the dipole operatorD and is a linear superposition of spherical harmonics

Ym~q!5 (l 5umu

`

~21! l Bl mYl m~q,0!. ~A1!

The coefficientsBl m are defined by the overlap integrals

Bl m5E d3x~Dc i !~x!A2/rJ2l 11~A8r !Yl m* ~q,w!

~A2!

@with Jn(x) Bessel functions# and can be calculated analytcally @8#. For excitations of the ground statec i5u1s0& withp-polarized light~i.e., dipole operatorD5z) the explicit re-sult is

Y0~q!52p21/223e22cosq ~A3!

and for c i5u2p0&; i.e., the initial state in many spectroscopic measurements on hydrogen@2,3,9# we obtain

Y0~q!5~2p!21/227e24~4cos2q21!. ~A4!

te

s

heas-

und

y


APPENDIX B: UNIFORM PHASE INTEGRALS

1. The Airy function Ai „x…

In the case of a fold catastrophe the uniform phase ingral is an Airy function@26#

E2`

1`

cos~ t31xt!dt5321/32pAi ~321/3x! ~B1!

with the asymptotic behavior

Ai ~6x! ;x→`H 1

2p21/2x21/4expS 2

2

3x3/2D

p21/2x21/4sinS 23 x3/21 p

4 D . ~B2!

2. The function F„x…

For a cusp catastrophe the uniform phase integral

F~x![E2`

1`

exp@2 i ~xt21t4!#dt ~B3!

is a special case of Pearcey’s integral@25,17,18#

P~x,y!5E2`

1`

exp@ i ~ t41xt21yt!#dt, ~B4!

i.e.,

F~x!5P* ~x,y50!. ~B5!

With the help of the integrals (p.q.0) @27#

E0

`

exp@2 i ~xp1txq!#dx

51

pexpF2 i

p

2pG (k50

`1

k!GS kq11

p D3S 2texpF i p2q

2pp G D k ~B6!

we obtain an absolutely convergent Taylor series forF(x)

F~x!51

2expF2 i

p

8 G (n50

`1

n!GS 2n11

4 D S xexpF2 i3

4pG D n.

~B7!

AlternativelyF(x) can be expressed in closed form in termof Bessel functions@27#

F~6x!5p

2Ax/2expF i x28 G

3H expF2 ip

8 GJ2 1/4S x28 D7expF i p

8 GJ1/4S x28 D J .~B8!

-

For largex we obtain asymptotic formulas by expanding tphases around their stationary points or by applying theymptotic formula

Jn~x! ;x→`

A~2/px!cosS x2pn

22

p

4 D ~B8!

of the Bessel functions to Eq.~B8!

F~6x!;

x→`H Ap/xexpF2 ip

4 GAp/xH expF i p

4 G1A2expF i S 14 x22 p

4 D G J .~B9!

3. The function C„x,y…

The uniform phase integralC(x,y) of the butterfly catas-trophe is expanded in a two-parametric Taylor series arox5y50

C~x,y![E2`

1`

exp@2 i ~xt21yt41t6!#dt

5 (n50

`

(m50

`1

i n1m

xnym

n!m! E2`

1`

t2n14mexp@2 i t 6#dt.

~B10!

With the substitutionz5t2n14m11 we obtain@27#

E2`

1`

t2n14mexp@2 i t 6#dt

52

2n14m11E0`

exp@2 iz6/~2n14m11!#dz

51

3expF2 i

2n14m11

12pGGS 2n14m11

6 D ,and finally

C~x,y!51

3expF2 i

p

12G (n50

`

(m50

`1

n!m!GS 2n14m11

6 D3S xexpF2 i

2

3p G D nS yexpF2 i

5

6p G Dm, ~B11!

which is a convergent series for allx andy.The asymptotic behavior ofC(x,y) for x→6` is ob-

tained in a stationary phase approximation to Eq.~B10!, withthe stationary pointst0 being defined by

t0~6t0414yt0

212x!50. ~B12!

~a! x→2`: There are three real stationary points given b

t050 and t0252

1

3y1S 19 y22 1

3xD 1/2,

and we obtain

neroxi--


C~x,y! ;x→2`

E2`

1`

exp@2 ixt2#dt

12exp@2 i ~xt021yt0

41t06!#

3E2`

1`

exp@2 i ~x16yt02115t0

4!t2#dt

5S p

2xD1/2

expF i p

4 G1F p

2x11

3y22yS 19 y22 1

3xD 1/2G 1/2

3expH i F2S 19 y22 1

3xD 3/22 1

3yS 29 y22xD2

p

4 G J .~B13!

~b! x→1`: The only real solution of Eq.~B12! is t050 andC(x,y) is approximated as

C~x,y! ;x→1`

E2`

1`

exp@2 ixt2#dt5S p

x D 1/2expF2 ip

4 G .~B14!

K.

e,.

y

li

os

tt..

In the case ofy,0 andx* 13 y

2 the complex zeros

t056F21

3y6 i ~ 1

3x2 19 y

2!1/2G1/2of Eq. ~B12! are situated close to the real axis. When oconsiders these complex zeros in a stationary phase appmation Eq.~B14! is modified by an additional term exponentially damped for largex

C~x,y! ;x→1`

S p

x D 1/2expF2 ip

4 G1F p

x21

3y22 iy~ 1

3x2 19 y

2!1/2G 1/23expF22S 13 x2

1

9y2D 3/2G

3expH 2 i F13 yS 29 y22xD2p

4 G J .~B15!

-

o,

l

,

@1# W. R. S. Garton and F. S. Tomkins, Astrophys. J.158, 839~1969!.

@2# A. Holle, G. Wiebusch, J. Main, B. Hager, H. Rottke, andH. Welge, Phys. Rev. Lett.56, 2594~1986!; J. Main, G. Wie-busch, A. Holle, and K. H. Welge,ibid. 57, 2789~1986!.

@3# A. Holle, J. Main, G. Wiebusch, H. Rottke, and K. H. WelgPhys. Rev. Lett.61, 161~1988!; J. Main, G. Wiebusch, and KH. Welge, Comments At. Mol. Phys.25, 233 ~1991!.

@4# H. Friedrich and D. Wintgen, Phys. Rep.183, 37 ~1989!.@5# H. Hasegawa, M. Robnik, and G. Wunner, Prog. Theor. Ph

Suppl.98, 198 ~1989!.@6# S. Watanabe, inReview of Fundamental Processes and App

cations of Atoms and Ions, edited by C. D. Lin~World Scien-tific, Singapore, 1993!.

@7# M. L. Du and J. B. Delos, Phys. Rev. A38, 1896~1988!; 38,1913 ~1988!.

@8# E. B. Bogomolny, Pis’ma Zh. E´ksp. Teor. Fiz.47, 445 ~1988!@JETP Lett.47, 526 ~1988!#; Zh. Eksp. Teor. Fiz.96, 487~1989! @Sov. Phys. JETP69, 275 ~1989!#.

@9# J. Main, G. Wiebusch, K. H. Welge, J. Shaw, and J. B. DelPhys. Rev. A49, 847 ~1994!.

@10# J. M. Mao and J. B. Delos, Phys. Rev. A45, 1746~1992!.@11# D. A. Sadovskiı´, J. A. Show, and J. B. Delos, Phys. Rev. Le

75, 2120~1995!; D. A. Sadovskiı´ and J. B. Delos, Phys. RevE 54, 2033~1996!.

@12# M. V. Berry and K. E. Mount, Rep. Prog. Phys.35, 315~1972!.

s.

-

,

@13# J. N. L. Connor, Mol. Phys.31, 33 ~1976!.@14# M. V. Berry and C. Upstill, inProgress in Optics, edited by E.

Wolf ~North-Holland, Amsterdam, 1980!, Vol. 18, p. 257–346.

@15# A. M. Ozorio de Almeida and J. H. Hannay, J. Phys. A20,5873 ~1987!.

@16# T. Poston and I. N. Steward,Catastrophe Theory and its Applications ~Pitman, London, 1978!.

@17# J. N. L. Connor, Mol. Phys.26, 1217~1973!.@18# J. N. L. Connor and D. Farrelly, J. Chem. Phys.75, 2831

~1981!.@19# A. D. Peters, C. Jaffe´, and J. B. Delos, Phys. Rev. Lett.73,

2825 ~1994!.@20# M. W. Beims and G. Alber, Phys. Rev. A48, 3123~1993!.@21# M. Courtney, Hong Jiao, N. Spellmeyer, D. Kleppner, J. Ga

and J. B. Delos, Phys. Rev. Lett.74, 1538~1995!.@22# M. Kus, F. Haake, and D. Delande, Phys. Rev. Lett.71, 2167

~1993!.@23# E. L. Stiefel and G. Scheifele,Linear and Regular Celestia

Mechanics~Springer, New York, 1971!.@24# B. Hupper, J. Main, and G. Wunner, Phys. Rev. Lett.74, 2650

~1995! and Phys. Rev. A53, 744 ~1996!.@25# T. Pearcey, Philos. Mag.37, 311 ~1946!.@26# Handbook of Mathematical Functions, edited by M.

Abramowitz and I. A. Stegun~Dover, New York, 1965!.@27# I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series

and Products~Academic, New York, 1965!.

hydrogen atom in a magnetic ﬁeld: ghost orbits, catastrophes, … · 2018. 6. 13. · axis, vm n...

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